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finished Quantifiers (again)

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wadler 2018-03-18 19:13:27 -03:00
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src/Quantifiers.lagda
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@ -61,21 +61,21 @@ is a term of type `A` then we may conclude that `B M` holds.
\end{code}
As with `→-elim`, the rule corresponds to function application.
Function types arise as a special case of dependent function types,
Functions arise as a special case of dependent functions,
where the range does not depend on a variable drawn from the domain.
When a function is viewed as evidence of implication, both its
argument and result are viewed as evidence, whereas when a dependent
function is viewed as evidence of a universal, its argument is viewed
as an element of a data type and its result is viewed as evidence of
a proposition that depends on the argument. This difference is largely
a matter of interpretation, since in Agda data types and types of
evidence are indistinguishable.
a matter of interpretation, since in Agda values of a type and
evidence of a proposition are indistinguishable.
Dependent function types are sometimes referred to as dependent products,
because if `A` is a finite type with values `{ x₁ , ⋯ , xᵢ }`, and if
each of the types `B x₁ , ⋯ B xᵢ` has `m₁ , ⋯ , mᵢ` distinct members,
then `∀ (x : A) → B x` has `m₁ × ⋯ × mᵢ` members. Because of this
association, sometimes the notation `∀ (x : A) → B x`
because if `A` is a finite type with values `x₁ , ⋯ , xᵢ`, and if
each of the types `B x₁ , ⋯ , B xᵢ` has `m₁ , ⋯ , mᵢ` distinct members,
then `∀ (x : A) → B x` has `m₁ * ⋯ * mᵢ` members.
Indeed, sometimes the notation `∀ (x : A) → B x`
is replaced by a notation such as `Π[ x ∈ A ] (B x)`,
where `Π` stands for product. However, we will stick with the name
dependent function, because (as we will see) dependent product is ambiguous.
@ -104,28 +104,41 @@ Does the converse hold? If so, prove; if not, explain why.
Given a variable `x` of type `A` and a proposition `B x` which
contains `x` as a free variable, the existentially quantified
proposition `∃ (λ (x : A) → B x)` holds if for some term `M` of type
proposition `Σ[ x ∈ A ] B x` holds if for some term `M` of type
`A` the proposition `B M` holds. Here `B M` stands for
the proposition `B x` with each free occurrence of `x` replaced by
`M`. The variable `x` appears free in `B x` but bound in
`∃ (λ (x : A) → B x)`.
`Σ[ x ∈ A ] B x`.
We formalise existential quantification by declaring a suitable
inductive type.
\begin{code}
data ∃ {A : Set} (B : A → Set) : Set where
_,_ : (x : A) → B x → ∃ B
data Σ (A : Set) (B : A → Set) : Set where
_,_ : (x : A) → B x → Σ A B
\end{code}
Evidence that `∃ (λ (x : A) → B x)` holds is of the form
We define a convenient syntax for existentials as follows.
\begin{code}
infix 2 Σ-syntax
Σ-syntax = Σ
syntax Σ-syntax A (λ x → B) = Σ[ x ∈ A ] B
\end{code}
This is our first use of a syntax declaration, which specifies that
the term on the left may be written with the syntax on the right.
The special syntax is available only when the identifier
`Σ-syntax` is imported.
Evidence that `Σ[ x ∈ A ] B x` holds is of the form
`(M , N)` where `M` is a term of type `A`, and `N` is evidence
that `B M` holds.
Equivalently, we could also declare existentials as a record type.
\begin{code}
record ∃′ {A : Set} (B : A → Set) : Set where
record Σ′ (A : Set) (B : A → Set) : Set where
field
proj₁ : A
proj₂ : B proj₁
proj₁′ : A
proj₂′ : B proj₁′
\end{code}
Here record construction
@ -140,65 +153,73 @@ corresponds to the term
where `M` is a term of type `A` and `N` is a term of type `B M`.
Given evidence that `∃ (λ (x : A) → B x)` holds, and
given evidence that `∀ (x : A) → B x → C` holds, where `C` does
not contain `x` as a free variable, we may conclude that `C` holds.
Products arise a special case of existentials, where the second
component does not depend on a variable drawn from the first
component. When a product is viewed as evidence of a conjunction,
both of its components are viewed as evidence, whereas when it is
viewed as evidence of an existential, the first component is viewed as
an element of a datatype and the second component is viewed as
evidence of a proposition that depends on the first component. This
difference is largely a matter of interpretation, since in Agda values
of a type and evidence of a proposition are indistinguishable.
Existentials are sometimes referred to as dependent sums,
because if `A` is a finite type with values `x₁ , ⋯ , xᵢ`, and if
each of the types `B x₁ , ⋯ B xᵢ` has `m₁ , ⋯ , mᵢ` distinct members,
then `Σ[ x ∈ A] B x` has `m₁ + ⋯ + mᵢ` members, which explains the
choice of notation for existentials, since `Σ` stands for sum.
Existentials are sometimes referred to as dependent products, since
products arise as a special case. However, that choice of names is
doubly confusing, since universal also have a claim to the name dependent
product and since existential also have a claim to the name dependent sum.
A common notation for existentials is `∃` (analogous to `∀` for universals).
We follow the convention of the Agda standard library, and reserve this
notation for the case where the domain of the bound variable is left implicit.
\begin{code}
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} →
(∃ (λ (x : A) → B x)) → (∀ (x : A) → B x → C) → C
∃-elim (M , N) P = P M N
∃ : ∀ {A : Set} (B : A → Set) → Set
∃ {A} B = Σ A B
∃-syntax = ∃
syntax ∃-syntax (λ x → B) = ∃[ x ] B
\end{code}
The special syntax is available only when the identifier `∃-syntax` is imported.
We will tend to use this syntax, since it is shorter and more familiar.
Given evidence that `∀ x → B x → C` holds, where `C` does not contain
`x` as a free variable, and given evidence that `∃[ x ] B x` holds, we
may conclude that `C` holds.
\begin{code}
∃-elim : ∀ {A : Set} {B : A → Set} {C : Set} → (∀ x → B x → C) → ∃[ x ] B x → C
∃-elim f (x , y) = f x y
\end{code}
In other words, if we know for every `x` of type `A` that `B x`
implies `C`, and we know for some `x` of type `A` that `B x` holds,
then we may conclude that `C` holds. This is because we may
instantiate that proof that `∀ (x : A) → B x → C` to any value
`M` of type `A` and any `N` of type `B M`, and exactly such values
are provided by the evidence for `∃ (λ (x : A) → B x)`.
instantiate that proof that `∀ x → B x → C` to any value `x` of type
`A` and any `y` of type `B x`, and exactly such values are provided by
the evidence for `∃[ x ] B x`.
Products arise a special case of existentials, where the second
component does not depend on a variable drawn from the first
component. When a product is viewed as evidence of a conjunction,
both of its components are viewed as evidence, whereas when viewed as
evidence of an existential, the first component is viewed as an
element of a datatype and the second component is viewed as evidence
of a proposition that depends on the first component. This difference
is largely a matter of interpretation, since in Agda data types and
types of evidence are indistinguishable.
Existentials are sometimes referred to as dependent sums,
because if `A` is a finite type with values `{ x₁ , ⋯ , xᵢ }`, and if
each of the types `B x₁ , ⋯ B xᵢ` has `m₁ , ⋯ , mᵢ` distinct members,
then `∃ (λ (x : A) → B x)` has `m₁ + ⋯ + mᵢ` members. Because of this
association, sometimes the notation `∃ (λ (x : A) → B x)`
is replaced by a notation such as `Σ[ x ∈ A ] (B x)`,
where `Σ` stands for sum.
Existentials are sometimes referred to as dependent products, since
products arise as a special case. However, that choice of names is
doubly confusing, because universals also have
a claim to the name dependent product, and because existentials
have a claim to the name dependent sum.
Agda makes it possible to define our own syntactic abbreviations.
Indeed, the converse also holds, and the two together form an isomorphism.
\begin{code}
∃-syntax = ∃
syntax ∃-syntax (λ x → B) = ∃[ x ] B
∀∃-currying : ∀ {A : Set} {B : A → Set} {C : Set} → (∀ x → B x → C) ≃ (∃[ x ] B x → C)
∀∃-currying =
record
{ to = λ{ f → λ{ (x , y) → f x y }}
; from = λ{ g → λ{ x → λ{ y → g (x , y) }}}
; from∘to = λ{ f → refl }
; to∘from = λ{ g → extensionality λ{ (x , y) → refl }}
}
\end{code}
This allows us to write `∃[ x ] (B x)` in place of `∃ (λ x → B x)`.
We could also define a syntax that makes the argument explicit.
\begin{code}
Σ-syntax = ∃
syntax Σ-syntax {A} (λ x → B) = Σ[ x ∈ A ] B
\end{code}
Both forms of syntax are provided by the Agda standard library.
We will usually use the first, as it is more compact.
The result can be viewed as a generalisation of currying, and the code to
establish the isomorphism is identical.
As an example, recall the definitions of `even` and `odd` from
Chapter [Relations](Relations).
## Existential examples
Recall the definitions of `even` and `odd` from Chapter [Relations](Relations).
\begin{code}
data even : ℕ → Set
data odd : ℕ → Set
@ -215,13 +236,15 @@ odd if it the successor of an even number.
We will show that a number is even if and only if it is twice some
other number, and odd if and only if it is one more than twice
some other number.
some other number. In other words, we will show
First, we need a lemma, which allows us to
simplify twice the successor of `m` to two
more than twice `m`.
> `even n` *iff* `∃[ m ] ( 2 * m ≡ n)`
> `odd n` *iff* `∃[ m ] (1 + 2 * m ≡ n)`
First, we need a lemma, which allows us to simplify twice the
successor of `m` to two more than twice `m`.
\begin{code}
lemma : ∀ (m : ℕ) → 2 * suc m ≡ suc (suc (2 * m))
lemma : ∀ (m : ℕ) → 2 * suc m ≡ 2 + 2 * m
lemma m =
begin
2 * suc m
@ -232,7 +255,7 @@ lemma m =
≡⟨ cong suc (+-suc m (m + zero)) ⟩
suc (suc (m + (m + zero)))
≡⟨⟩
suc (suc (2 * m))
2 + 2 * m
∎
\end{code}
The lemma is straightforward, and uses the lemma
@ -367,7 +390,7 @@ Does the converse hold? If so, prove; if not, explain why.
Definitions similar to those in this chapter can be found in the standard library.
\begin{code}
import Data.Product using (∃;_,_)
import Data.Product using (Σ; _,_; ∃; Σ-syntax; ∃-syntax)
\end{code}